In this paper, we derive \(L_2\)  concentration inequalities for quadratic forms in symmetric random tensors with optimal dependence on \(d\)  (at least, when \(d=O(n)\) ). Our results improve those of [1]} and have some relevance to random chaoses. However, as noted in [1]}, known concentration inequalities for random chaoses [3]}, [4]}, [5]}, [6]}, [7]}, [8]} exhibit an unspecified (possibly exponential) dependence on the degree \(d\) , which is too bad for our problem. Also, comparing to the non-symmetric case, the strong dependence structure of the symmetric random tensor model highly complicates its analysis. In particular,
for the quadratic form given by the squared \(l_2\)  norm, \(\Vert X^{(1)}\otimes \ldots \otimes X^{(d)}\Vert ^2\)  is the product \(\prod _{k=1}^d\Vert X^{(k)}\Vert ^2\)  of independent random variables and \(\Vert X^{\otimes d}\Vert ^2\)  is the elementary symmetric polynomial of order \(d\)  in the squared entries of \(X\) . The latter plays a key role in the context of the MP theorem for the symmetric random tensor model, as the necessary condition for the theorem is given by \(\binom{n}{d}^{-1}\Vert X^{\otimes d}\Vert ^2\rightarrow 1\)  in probability when \(X=X(n)\) , \(d=d(n)\) , and \(n\rightarrow \infty \)  (see Section ). This condition could be viewed as a law of large numbers for elementary symmetric polynomials in independent nonnegative random variables. The asymptotic behaviour of such polynomials is thoroughly studied in [9]}, [10]}, [11]}, [12]}. Using the saddle-point approximation method of [10]}, we find necessary and sufficient conditions (in terms of \(d\)  and \(n\) ) for the above law of large numbers in the i.i.d. case.
